The element carbon plays a central role in climate and life on Earth. It is capable of moving among the geosphere, cryosphere, atmosphere, biosphere and hydrosphere. This flow of carbon is referred to as the Earth's carbon cycle. It is also intimately linked to the cycling of other elements and compounds. The ocean plays a fundamental role in Earth's carbon cycle, helping to regulate atmospheric CO₂ concentration. The ocean biological carbon pump (OBCP), defined as a set of processes that transfer organic carbon from the surface to the deep ocean, is at the heart of the ocean carbon cycle. Monitoring the OBCP is critical to understanding how the Earth's carbon cycle is changing. At present, satellite remote sensing is the only tool available for viewing the entire surface ocean at high temporal and spatial scales. In this paper, we review methods for monitoring the OBCP with a focus on satellites. We begin by providing an overview of the OBCP, defining and describing the pools of carbon in the ocean, and the processes controlling fluxes of carbon between the pools, from the surface to the deep ocean, and among ocean, land and atmosphere. We then examine how field measurements, from ship and autonomous platforms, complement satellite observations, provide validation points for satellite products and lead to a more complete view of the OBCP than would be possible from satellite observations alone. A thorough analysis is then provided on methods used for monitoring the OBCP from satellite platforms, covering current capabilities, concepts and gaps, and the requirement for uncertainties in satellite products. We finish by discussing the potential for producing a satellite-based carbon budget for the oceans, the advantages of integrating satellite-based observations with ecosystem models and field measurements, and future opportunities in space, all with a view towards bringing satellite observations into the limelight of ocean carbon research.